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In this paper we design protograph-based LDPC codes with short block sizes. Mainly we consider rate 1/2 codes with input block sizes 64, 128, and 256 bits. To simplify the encoder and decoder implementations for high data rate transmission, the structure of the codes is based on protographs and circulants. These codes are designed for short block sizes based on maximizing the minimum distance and stopping set size subject to a constraint on the maximum variable node degree. In particular, we consider codes with variable node degrees between 3 and 5. Increasing the node degree leads to larger minimum distances, at the expense of smaller girth. Therefore, there is a trade-off between undetected error rate performance (improved by increasing minimum distance) and the degree of sub-optimality of the iterative decoders typically used (which are adversely affected by graph loops). Various LDPC codes are compared and simulation results are provided.